Depends on what jargon you're using. I've always taken "manufactured housing" to mean essentially a house trailer. Now that can be a single-wide, a double-wide, or even a triple-wide. And it can be be put on a permanent foundation, or even have a roof with a slight pitch to it. But if it came in on its own frame and own set of wheels, and could also be put up on cinder blocks, then it's a "manufactured home".
I've heard the phrase "Mobile Home" came into common usage because one of the earliest manufacturers of them was located near Mobile Alabama. And in later years people thought of "Mobile" as in, a trailer you can move about as you please, although these things are really only designed to be moved once, towed by a heavy-duty truck, and then blocked and skirted, either in a trailer park or private lot. They're really not sturdy enough to be moved about regularly, though.
"Modular Home" on the other hand, is what I've taken to mean a home that is built in sections, in a factory, and then brought out and set into place on a permanent foundation. They can be split foyer, split level, two story, or a Cape Cod, any design, actually. And once put together and finished off, they don't look any different from a stick-built home.
My Mom and stepdad live in a modular home. It's a rather modest 24x48 rancher, built in 1979. The only tip-off that it's a modular home is on the inside. The bearing-wall that runs down the center is extra thick, because each side of the 12x48' sections was designed to be self-supporting. The roof was also site-built, and separate, whereas on a manufactured/mobile home, the ones that have a slight roof pitch have the roof actually built into the structure, I believe, and then when the two pieces are joined together on the side, the top layer of shingles/ridge vent, etc are installed to finish it off.
Most manufactured/Mobile homes, even the nicer ones, always give themselves away to me, because the outer walls are usually only 7'6" tall, rather than your typical 8', or more. The inner walls are usually thinner as well. And often, rather than having what I'd call a "fully-finished" wall (drywalled, spackled, and painted), they just have these 4x8 panels that have thin strips at the seams, sort of like an RV.
I'm not trying to rag on manufactured/Mobile homes, though. I've seen some of them that are actually quite nice. I have some relatives that have owned them, and as far as I know, never had problems with them
I do remember one issue that my cousins had, though. They had bought a large parcel of land in Southern Maryland, on a fairly new street, and wanted to put in a manufactured home. I think it was built by Fleetwood, and 28x76 feet. Some of the neighbors, who had already had stick-built houses put in, did raise a fuss, although I don't think there was any kind of HOA they could have griped to. But, my cousins were able to get it put on the land, anyway, and apparently once it was finished, the neighbors chilled out.
